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Mirrors > Home > MPE Home > Th. List > nfielex | Structured version Visualization version GIF version |
Description: If a class is not finite, then it contains at least one element. (Contributed by Alexander van der Vekens, 12-Jan-2018.) |
Ref | Expression |
---|---|
nfielex | ⊢ (¬ 𝐴 ∈ Fin → ∃𝑥 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0fin 9170 | . . . 4 ⊢ ∅ ∈ Fin | |
2 | eleq1 2815 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ∈ Fin ↔ ∅ ∈ Fin)) | |
3 | 1, 2 | mpbiri 258 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin) |
4 | 3 | con3i 154 | . 2 ⊢ (¬ 𝐴 ∈ Fin → ¬ 𝐴 = ∅) |
5 | neq0 4340 | . 2 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
6 | 4, 5 | sylib 217 | 1 ⊢ (¬ 𝐴 ∈ Fin → ∃𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∅c0 4317 Fincfn 8938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-om 7852 df-en 8939 df-fin 8942 |
This theorem is referenced by: cusgrfi 29219 esumcst 33590 topdifinffinlem 36734 |
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